Sufficiently Generic Orthogonal Grassmannians
نویسنده
چکیده
We prove the following conjecture due to Bryant Mathews (2008). Let Qi be the orthogonal grassmannian of totally isotropic i-planes of a non-degenerate quadratic form q over an arbitrary field (where i is an integer satisfying 1 ≤ i ≤ m := [(dim q)/2]). Assume that for a given i, the form q has the following property (possessed by the generic quadratic form): the degree of each closed point on Qi is divisible by 2 i and the Witt index of q over the function field of Qi is equal to i. Then the variety Qi is 2-incompressible. Assuming that the form q is sufficiently close to the generic one in a different sense, we prove a stronger property of Qi saying that its Chow motive with coefficients in F2 (the finite field of 2 elements) is indecomposable. This result contrasts with recent results of Zhykhovich (2010) [21] on decomposability of the motives of incompressible twisted
منابع مشابه
Unitary Grassmannians
We study projective homogeneous varieties under an action of a projective unitary group (of outer type). We are especially interested in the case of (unitary) grassmannians of totally isotropic subspaces of a hermitian form over a field, the main result saying that these grassmannians are 2-incompressible if the hermitian form is generic. Applications to orthogonal grassmannians are provided.
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